In [1]}, [2]} a different approach to the study of (REF ) is introduced. This approach is based on the construction of a suitable self adjoint realization of \( \Delta + \xi \) .
Then the initial data in (REF ) is chosen in the domain of this self adjoint operator. Being in the domain of this self adjoint operator is the substitute of our assumption of well prepared initial data \(e^{Y(x,\omega )} u_0(x) \in H^2(\mathbb {T}^2)\) . At the best of our knowledge the present paper is the first one where
global well-posedness is proved for (REF ) with an arbitrary polynomial nonlinearity \(p\) , extending the papers [3]} and [4]}. Our proof is based on the approach
introduced by Hairer-Labbé in [5]}.
